Theorem xrmaxleim 10852

Description: Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.)

Assertion

Ref	Expression
xrmaxleim	|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B -> sup ( { A , B } , RR* , < ) = B ) )

Proof of Theorem xrmaxleim

Dummy variables f g y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrlttri3 9424	. . . 4 |- ( ( f e. RR* /\ g e. RR* ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
2	1	adantl 273	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ ( f e. RR* /\ g e. RR* ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
3		simplr 500	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) -> B e. RR* )
4		prid2g 3575	. . . 4 |- ( B e. RR* -> B e. { A , B } )
5	3, 4	syl 14	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) -> B e. { A , B } )
6		simpllr 504	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> A <_ B )
7		xrlenlt 7701	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) )
8	7	ad3antrrr 479	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> ( A <_ B <-> -. B < A ) )
9	6, 8	mpbid 146	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> -. B < A )
10		breq2 3879	. . . . . . 7 |- ( y = A -> ( B < y <-> B < A ) )
11	10	notbid 633	. . . . . 6 |- ( y = A -> ( -. B < y <-> -. B < A ) )
12	11	adantl 273	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> ( -. B < y <-> -. B < A ) )
13	9, 12	mpbird 166	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> -. B < y )
14		xrltnr 9407	. . . . . 6 |- ( B e. RR* -> -. B < B )
15	14	ad4antlr 482	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = B ) -> -. B < B )
16		breq2 3879	. . . . . . 7 |- ( y = B -> ( B < y <-> B < B ) )
17	16	notbid 633	. . . . . 6 |- ( y = B -> ( -. B < y <-> -. B < B ) )
18	17	adantl 273	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = B ) -> ( -. B < y <-> -. B < B ) )
19	15, 18	mpbird 166	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = B ) -> -. B < y )
20		elpri 3497	. . . . 5 |- ( y e. { A , B } -> ( y = A \/ y = B ) )
21	20	adantl 273	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) -> ( y = A \/ y = B ) )
22	13, 19, 21	mpjaodan 753	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) -> -. B < y )
23	2, 3, 5, 22	supmaxti 6806	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) -> sup ( { A , B } , RR* , < ) = B )
24	23	ex 114	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B -> sup ( { A , B } , RR* , < ) = B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 670   = wceq 1299   e. wcel 1448   {cpr 3475   class class class wbr 3875   supcsup 6784   RR*cxr 7671   < clt 7672   <_ cle 7673

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-pre-ltirr 7607  ax-pre-apti 7610

This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-xp 4483  df-cnv 4485  df-iota 5024  df-riota 5662  df-sup 6786  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678

This theorem is referenced by:  xrmaxltsup  10866  xrmaxadd  10869  xrmineqinf  10877